A strongly semismooth integral function and its application

Liqun Qi, Hongxia Yin

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)


As shown by an example, the integral function f : ℝn→ ℝ, defined by f(x) = ∫ab[B(x, t)]+g(t)dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ℝn. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g ≢ 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.
Original languageEnglish
Pages (from-to)223-246
Number of pages24
JournalComputational Optimization and Applications
Issue number1-3
Publication statusPublished - 1 Apr 2003


  • Generalized Newton method
  • Integral function
  • Piecewise smoothness
  • Quadratic convergence
  • Strong semismoothness

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


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